In the equation E = mc^2 we are referring to a specific particle with mass m. Also, dark energy is not an "energy" and dark matter is not a "mass." You simply can't connect the two concepts like this. Dark matter and dark energy have no relation to each other.

Yep. As Topquark says, the two phenomena are completely unrelated to each other.

The reason for the moniker "dark" is a historical naming artifact because in astrophysics almost everything we know about celestial objects is the result of the analysis of light we receive from those celestial objects. Because those phenomena (specifically, the flat rotation profiles in galaxies and the accelerating expansion of the observable universe) are unknown, they are often referred to as "dark".

Note that in other fields of study, the naming of unknown phenomena follows other schemes, more as a matter of preserving traditions. As an example, we can consider particle physics. Unknown particles are often given an algebraic letter (e.g. x-rays), given a name after the Greek letters used in the theoretical papers where they were predicted (e.g. pions, muons, tauons, kaons, etc.) or a description (up, down, top, bottom, charm and strange quarks).